4-4 Johannes Kepler proposed elliptical paths for the planets about the Sun

Kepler’s ideas apply not just to the planets, but to all orbiting celestial objects

The task that Johannes Kepler took on at the beginning of the seventeenth century was to find a model of planetary motion that agreed completely with Tycho’s extensive and very accurate observations of planetary positions. To do this, Kepler found that he had to break with an ancient prejudice about planetary motions.

Elliptical Orbits and Kepler’s First Law

Astronomers had long assumed that heavenly objects move in circles, which were considered the most perfect and harmonious of all geometric shapes. They believed that if a perfect God resided in heaven along with the stars and planets, then the motions of these objects must be perfect too. Against this context, Kepler dared to try to explain planetary motions with noncircular curves. In particular, he found that he had the best success with a particular kind of curve called an ellipse.

79

You can draw an ellipse by using a loop of string, two thumbtacks, and a pencil, as shown in Figure 4-10a. Each thumbtack in the figure is at a focus (plural foci) of the ellipse; an ellipse has two foci. The longest diameter of an ellipse, called the major axis, passes through both foci. Half of that distance is called the semimajor axis and is usually designated by the letter a. A circle is a special case of an ellipse in which the two foci are at the same point (this corresponds to using only a single thumbtack in Figure 4-10b). The semimajor axis of a circle is equal to its radius.

Figure 4-10: Ellipses (a) To draw an ellipse, use two thumbtacks to secure the ends of a piece of string, then use a pencil to pull the string taut. If you move the pencil while keeping the string taut, the pencil traces out an ellipse. The thumbtacks are located at the two foci of the ellipse. The major axis is the greatest distance across the ellipse; the semimajor axis is half of this distance. (b) A series of ellipses with the same major axis but different eccentricities. An ellipse can have any eccentricity from e = 0 (a circle) to just under e = 1 (virtually a straight line).

By assuming that planetary orbits were ellipses, Kepler found, to his delight, that he could make his theoretical calculations match precisely to Tycho’s observations. This important discovery, first published in 1609, is now called Kepler’s first law:

The orbit of a planet about the Sun is an ellipse with the Sun at one focus.

The semimajor axis a of a planet’s orbit is the average distance between the planet and the Sun.

CAUTION!

The Sun is at one focus of a planet’s elliptical orbit, but there is nothing at the other focus. This “empty focus” has geometrical significance, because it helps to define the shape of the ellipse, but plays no other role.

Ellipses come in different shapes, depending on the elongation of the ellipse. The shape of an ellipse is described by its eccentricity, designated by the letter e. Figure 4-10b shows a few examples of ellipses with different eccentricities. The value of e can range from 0 (a circle) to just under 1 (nearly a straight line). The greater the eccentricity, the more elongated the ellipse. Because a circle is a special case of an ellipse, it is possible to have a perfectly circular orbit. But all of the objects that orbit the Sun have orbits that are at least slightly elliptical. The most circular of any planetary orbit is that of Venus, with an eccentricity of just 0.007; Mercury’s orbit has an eccentricity of 0.206, and a number of small bodies called comets move in very elongated orbits with eccentricities of less than 1.

CONCEPT CHECK 4-9

Which orbit is closer to being circular—Venus’s orbit, with e = 0.007, or Mars’s orbit, with e = 0.093?

80

Orbital Speeds and Kepler’s Second Law

Once he knew the shape of a planet’s orbit, Kepler was ready to describe exactly how it moves on that orbit. As a planet travels in an elliptical orbit, its distance from the Sun varies. Kepler realized that the speed of a planet also varies along its orbit. A planet moves most rapidly when it is nearest the Sun, at a point on its orbit called perihelion. Conversely, a planet moves most slowly when it is farthest from the Sun, at a point called aphelion (Figure 4-11). After much trial and error, Kepler found a way to describe just how a planet’s speed varies as it moves along its orbit. Figure 4-11 illustrates this discovery, referred to as Kepler’s second law:

A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

Figure 4-11: Kepler’s First and Second Laws According to Kepler’s first law, a planet travels around the Sun along an elliptical orbit with the Sun at one focus. According to his second law, a planet moves fastest when closest to the Sun (at perihelion) and slowest when farthest from the Sun (at aphelion). As the planet moves, an imaginary line joining the planet and the Sun sweeps out equal areas in equal intervals of time (from A to B or from C to D). By using these laws in his calculations, Kepler found an excellent fit to the apparent motions of the planets.

This relationship is also called the law of equal areas. In the idealized case of a circular orbit, a planet would have to move at a constant speed around the orbit in order to satisfy Kepler’s second law. In the case of an elliptical orbit, a planet speeds up when it is closer to the Sun, and slows down when it is farther away.

ANALOGY

An analogy for Kepler’s second law is a twirling ice skater holding weights in each hand. If the skater moves the weights closer to her body by pulling her arms straight in, her rate of spin increases and the weights move faster; if she extends her arms so the weights move away from her body, her rate of spin decreases and the weights slow down. Just like the weights, a planet in an elliptical orbit travels at a higher speed when it moves closer to the Sun (toward perihelion) and travels at a lower speed when it moves away from the Sun (toward aphelion).

CONCEPT CHECK 4-10

According to Kepler’s second law, at what point in a communications satellite’s elliptical orbit around Earth will it move the slowest?

Orbital Periods and Kepler’s Third Law

Kepler’s second law describes how the speed of a given planet changes as it orbits the Sun. Kepler also deduced from Tycho’s data a relationship that can be used to compare the motions of different planets. Published in 1618 and now called Kepler’s third law, it states a relationship between the size of a planet’s orbit and the time the planet takes to go once around the Sun:

The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit.

Kepler’s third law says that the larger a planet’s orbit—that is, the larger the semimajor axis, or average distance from the planet to the Sun—the longer the sidereal period, which is the time it takes the planet to complete an orbit. From Kepler’s third law one can show that the larger the semimajor axis, the slower the average speed at which the planet moves around its orbit. (By contrast, Kepler’s second law describes how the speed of a given planet is sometimes faster and sometimes slower than its average speed.) This qualitative relationship between orbital size and orbital speed is just what Aristarchus and Copernicus used to explain retrograde motion, as we saw in Section 4-2. Kepler’s great contribution was to make this relationship a quantitative one.

81

It is useful to restate Kepler’s third law as an equation. If a planet’s sidereal period P is measured in years and the length of its semimajor axis a is measured in astronomical units (AU), where 1 AU is the average distance from Earth to the Sun (see Section 1-7), then Kepler’s third law is

Kepler’s third law

P2 = a3

If you know either the sidereal period of a planet or the semimajor axis of its orbit, you can find the other quantity using this equation. Box 4-2 gives some examples of how this is done.

We can verify Kepler’s third law for all of the planets, including those that were discovered after Kepler’s death, using data from Table 4-1 and Table 4-2. If Kepler’s third law is correct, for each planet the numerical values of P2 and a3 should be equal. This result is indeed true to very high accuracy, as Table 4-3 shows.

Table 4-3: A Demonstration of Kepler’s Third Law (P2 = a3)
Planet Sidereal period P (years) Semimajor axis a (AU) P2 a3
Mercury     0.24     0.39     0.06     0.06
Venus     0.61     0.72     0.37     0.37
Earth     1.00     1.00     1.00     1.00
Mars     1.88     1.52     3.53     3.51
Jupiter   11.86     5.20   140.7   140.6
Saturn   29.46     9.55   867.9   871.0
Uranus   84.10   19.19   7,072   7,067
Neptune 164.86   30.07 27,180 27,190
Kepler’s third law states that P2 = a3 for each of the planets. The last two columns of this table demonstrate that this relationship holds true to a very high level of accuracy.

TOOLS OF THE ASTRONOMER’S TRADE

Using Kepler’s Third Law

Kepler’s third law relates the sidereal period P of an object orbiting the Sun to the semimajor axis a of its orbit:

P2 = a3

You must keep two essential points in mind when working with this equation:

  1. The period P must be measured in years, and the semimajor axis a must be measured in astronomical units (AU). Otherwise you will get nonsensical results.
  2. This equation applies only to the special case of an object, like a planet, that orbits the Sun. If you want to analyze the orbit of the Moon around Earth, of a spacecraft around Mars, or of a planet around a distant star, you must use a different, generalized form of Kepler’s third law. We discuss this alternative equation in Section 4-7 and Box 4-4.

EXAMPLE: The average distance from Venus to the Sun is 0.72 AU. Use this to determine the sidereal period of Venus.

Situation: The average distance from Venus to the Sun is the semimajor axis a of the planet’s orbit. Our goal is to calculate the planet’s sidereal period P.

Tools: To relate a and P we use Kepler’s third law, P2 = a3.

Answer: We first cube the semimajor axis (multiply it by itself twice):

a3 = (0.72)3 = 0.72 × 0.72 × 0.72 = 0.373

According to Kepler’s third law this is also equal to P2, the square of the sidereal period. So, to find P, we have to “undo” the square, that is, take the square root. Using a calculator, we find

Review: The sidereal period of Venus is 0.61 years, or a bit more than seven Earth months. This result makes sense: A planet with a smaller orbit than Earth’s (an inferior planet) must have a shorter sidereal period than Earth.

EXAMPLE: A certain small asteroid (a rocky body a few tens of kilometers across) takes eight years to complete one orbit around the Sun. Find the semimajor axis of the asteroid’s orbit.

Situation: We are given the sidereal period P = 8 years, and are to determine the semimajor axis a.

Tools: As in the preceding example, we relate a and P using Kepler’s third law, P2 = a3.

Answer: We first square the period:

P2 = 82 = 8 × 8 = 64

From Kepler’s third law, 64 is also equal to a3. To determine a, we must take the cube root of a3, that is, find the number whose cube is 64. If your calculator has a cube root function, denoted by the symbol , you can use it to find that the cube root of 64 is 4: = 4. Otherwise, you can determine by trial and error that the cube of 4 is 64:

43 = 4 × 4 × 4 = 64

Because the cube of 4 is 64, it follows that the cube root of 64 is 4 (taking the cube root “undoes” the cube).

With either technique you find that the orbit of this asteroid has semimajor axis a = 4 AU.

Review: The period is greater than 1 year, so the semimajor axis is greater than 1 AU. Note that a = 4 AU is intermediate between the orbits of Mars and Jupiter (see Table 4-3). Many asteroids are known with semimajor axes in this range, forming a region in the solar system called the asteroid belt.

82

CONCEPT CHECK 4-11

The space shuttle typically orbited Earth at an altitude of 300 km whereas the International Space Station orbits Earth at an altitude of 450 km. Although the space shuttle took less time to orbit Earth, which orbiter moved at a faster speed?

CALCULATION CHECK 4-1

If Pluto’s orbit has a semimajor axis of 39.5 AU, how long does it take Pluto to orbit the Sun once?

The Significance of Kepler’s Laws

Kepler’s laws are a landmark in the history of astronomy. They made it possible to calculate the motions of the planets with better accuracy than any geocentric model ever had, and they helped to justify the idea of a heliocentric model. Kepler’s laws also pass the test of Occam’s razor, for they are simpler in every way than the schemes of Ptolemy or Copernicus, both of which used a complicated combination of circles.

But the significance of Kepler’s laws goes beyond understanding planetary orbits. These same laws are also obeyed by spacecraft orbiting Earth, by two stars revolving about each other in a binary star system, and even by galaxies in their orbits about each other. Throughout this book, we shall use Kepler’s laws in a wide range of situations.

As impressive as Kepler’s accomplishments were, he did not prove that the planets orbit the Sun, nor was he able to explain why planets move in accordance with his three laws. These advances were made by two other figures who loom large in the history of astronomy: Galileo Galilei and Isaac Newton.

CONCEPT CHECK 4-12

Do Kepler’s laws of planetary motion apply only to the planets?